https://flutter.dev/docs/cookbook/networking/web-sockets use this tutorial. Also look into the concept of sockets. Keep in mind when you making a chat application you should also take into account privacy and security.

1. Connect to a WebSocket server.

   The web_socket_channel package provides the tools you need to connect to a WebSocket server. The package provides a WebSocketChannel that allows you to both listen for messages from the server and push messages to the server.

   In Flutter, use the following line to create a WebSocketChannel that connects to a server:

   final channel = WebSocketChannel.connect(
     Uri.parse('wss://echo.websocket.org'),
   );
   

2. Listen for messages from the server.

   Now that you've established a connection, listen to messages from the server. After sending a message to the test server, it sends the same message back. In this example, use a StreamBuilder widget to listen for new messages, and a Text widget to display them.

   StreamBuilder(
     stream: channel.stream,
     builder: (context, snapshot) {
       return Text(snapshot.hasData ? '${snapshot.data}' : '');
     },    
   )
   

3. Send data to the server.

   To send data to the server, add() messages to the sink provided by the WebSocketChannel.

   channel.sink.add('Hello!');
   

4. Close the WebSocket connection.

   After you're done using the WebSocket, close the connection:

   channel.sink.close();
   

Check out Askless, it simplifies the creation of real-time Flutter Apps using any database

I developed this package with the goal of simplifying the websocket usage. The library will automatically send again the data to the server if the client loses a connection, also it has other benefits. The Askless server side needs to be coded in JavaScript/TypeScript.

The Flutter Client will be able to listen to real-time updates using MySQL or other databases, like:

//other widgets...

AsklessClient.instance
    .listenAndBuild(
      route: 'allProducts',
      builder: (context,  snapshot) {
          if(!snapshot.hasData)
             return Container();

          final listOfProductsNames =
              (snapshot.data as List)
              .map((product) => Text(product['name'])).toList();

          return Column(
             children: listOfProductsNames,
          );
      }
    ),

//other widgets...

It also has a JavaScript/TypeScript client support that can be paired with Flutter if, for example, there is a web version of your App using React or other SPA framework.